Zubov's method is a technique for computing the basin of attraction for a set of ordinary differential equations (a dynamical system). The domain of attraction is the set { x : v ( x ) < 1 } {\displaystyle \{x:\,v(x)<1\}} , where v ( x ) {\displaystyle v(x)} is the solution to a partial differential equation known as the Zubov equation. Zubov's method can be used in a number of ways.
Statement Zubov's theorem states that:
If x ′ = f ( x ) , t ∈ R {\displaystyle x'=f(x),t\in \mathbb {R} } is an ordinary differential equation in R n {\displaystyle \mathbb {R} ^{n}} with f ( 0 ) = 0 {\displaystyle f(0)=0} , a set A {\displaystyle A} containing 0 in its interior is the domain of attraction of zero if and only if there exist continuous functions v , h {\displaystyle v,h} such that:
v ( 0 ) = h ( 0 ) = 0 {\displaystyle v(0)=h(0)=0} , 0 < v ( x ) < 1 {\displaystyle 0<v(x)<1} for x ∈ A ∖ { 0 } {\displaystyle x\in A\setminus \{0\}} , h > 0 {\displaystyle h>0} on R n ∖ { 0 } {\displaystyle \mathbb {R} ^{n}\setminus \{0\}}
for every γ 2 > 0 {\displaystyle \gamma _{2}>0} there exist γ 1 > 0 , α 1 > 0 {\displaystyle \gamma _{1}>0,\alpha _{1}>0} such that v ( x ) > γ 1 , h ( x ) > α 1 {\displaystyle v(x)>\gamma _{1},h(x)>\alpha _{1}} , if | | x | | > γ 2 {\displaystyle ||x||>\gamma _{2}}
v ( x n ) → 1 {\displaystyle v(x_{n})\rightarrow 1} for x n → ∂ A {\displaystyle x_{n}\rightarrow \partial A} or | | x n | | → ∞ {\displaystyle ||x_{n}||\rightarrow \infty }
∇ v ( x ) ⋅ f ( x ) = − h ( x ) ( 1 − v ( x ) ) 1 + | | f ( x ) | | 2 {\displaystyle \nabla v(x)\cdot f(x)=-h(x)(1-v(x)){\sqrt {1+||f(x)||^{2}}}}
If f is continuously differentiable, then the differential equation has at most one continuously differentiable solution satisfying v ( 0 ) = 0 {\displaystyle v(0)=0} .
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